A328596 -id:A328596 - cp's OEIS frontend

A329324 Number of Lyndon compositions of n whose reverse is not a co-Lyndon composition.

0, 0, 0, 0, 0, 1, 2, 7, 16, 37, 76, 166, 328, 669, 1326, 2626, 5138, 10104, 19680, 38442, 74822, 145715, 283424, 551721, 1073224

Offset: 1

Gus Wiseman, Nov 11 2019

A Lyndon composition of n is a finite sequence summing to n that is lexicographically strictly less than all of its cyclic rotations. A co-Lyndon composition of n is a finite sequence summing to n that is lexicographically strictly greater than all of its cyclic rotations.

			The a(6) = 1 through a(9) = 16 compositions:
  (132)  (142)   (143)    (153)
         (1132)  (152)    (162)
                 (1142)   (243)
                 (1232)   (1143)
                 (1322)   (1152)
                 (11132)  (1242)
                 (11312)  (1332)
                          (1422)
                          (11142)
                          (11232)
                          (11322)
                          (11412)
                          (12132)
                          (111132)
                          (111312)
                          (112212)

Lyndon and co-Lyndon compositions are counted by A059966.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Lyndon compositions that are not weakly increasing are A329141.
Cf. A000740, A001037, A008965, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
colynQ[q_]:=Array[Union[{RotateRight[q,#1],q}]=={RotateRight[q,#1],q}&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],lynQ[#]&&!colynQ[Reverse[#]]&]],{n,15}]

a(21)-a(25) from Robert Price, Jun 20 2021

A329325 Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n with first digit removed.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 1, 4, 2, 1, 1, 2, 2, 3, 1, 4, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 1, 5, 3, 1, 1, 5, 4, 1, 5, 2, 1

Offset: 1

Gus Wiseman, Nov 11 2019

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

			Triangle begins:
   1: ()        21: (22)       41: (23)       61: (1112)
   2: (1)       22: (31)       42: (221)      62: (11111)
   3: (1)       23: (4)        43: (5)        63: (11111)
   4: (11)      24: (1111)     44: (311)      64: (111111)
   5: (2)       25: (13)       45: (32)       65: (6)
   6: (11)      26: (121)      46: (41)       66: (51)
   7: (11)      27: (13)       47: (5)        67: (6)
   8: (111)     28: (1111)     48: (11111)    68: (411)
   9: (3)       29: (112)      49: (14)       69: (6)
  10: (21)      30: (1111)     50: (131)      70: (51)
  11: (3)       31: (1111)     51: (14)       71: (6)
  12: (111)     32: (11111)    52: (1211)     72: (3111)
  13: (12)      33: (5)        53: (122)      73: (33)
  14: (111)     34: (41)       54: (131)      74: (51)
  15: (111)     35: (5)        55: (14)       75: (6)
  16: (1111)    36: (311)      56: (11111)    76: (411)
  17: (4)       37: (5)        57: (113)      77: (6)
  18: (31)      38: (41)       58: (1121)     78: (51)
  19: (4)       39: (5)        59: (113)      79: (6)
  20: (211)     40: (2111)     60: (11111)    80: (21111)
For example, the trimmed binary expansion of 41 is (01001), with Lyndon factorization (01)(001), so row 41 is {2,3}.

Row lengths are A211097.
Row sums are A000523.
Keeping the first digit gives A329314.
Positions of singleton rows are A329327.
Binary Lyndon words are counted by A001037 and ranked by A102659.
Numbers whose reversed binary expansion is a Lyndon word are A328596.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A059966, A211097, A257250, A275692, A296372, A328594, A329313, A329315, A329318.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#1]]&]]]];
Table[Length/@lynfac[Rest[IntegerDigits[n,2]]],{n,100}]

A333765 Number of co-Lyndon factorizations of the k-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 4, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 2, 4, 4, 7, 7, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 5, 2, 5, 2, 4, 4, 9, 4, 7, 7, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 4, 1

Offset: 0

Gus Wiseman, Apr 13 2020

nonn

We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). A co-Lyndon factorization of a composition c is a multiset of compositions whose co-Lyndon product is c.
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
Also the number of multiset partitions of the co-Lyndon-word factorization of the n-th composition in standard order.

			The a(54) = 5, a(61) = 7, and a(237) = 9 factorizations:
  ((1,2,1,2))      ((1,1,1,2,1))        ((1,1,2,1,2,1))
  ((1),(2,1,2))    ((1),(1,1,2,1))      ((1),(1,2,1,2,1))
  ((1,2),(2,1))    ((1,1),(1,2,1))      ((1,1),(2,1,2,1))
  ((2),(1,2,1))    ((2,1),(1,1,1))      ((1,2,1),(1,2,1))
  ((1),(2),(2,1))  ((1),(1),(1,2,1))    ((2,1),(1,1,2,1))
                   ((1),(1,1),(2,1))    ((1),(1),(2,1,2,1))
                   ((1),(1),(1),(2,1))  ((1,1),(2,1),(2,1))
                                        ((1),(2,1),(1,2,1))
                                        ((1),(1),(2,1),(2,1))

The dual version is A333940.
Binary necklaces are counted by A000031.
Necklace compositions are counted by A008965.
Necklaces covering an initial interval are counted by A019536.
Lyndon compositions are counted by A059966.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a necklace are A329138.
Length of Lyndon factorization of binary expansion is A211100.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of co-Lyndon factorization of reversed binary expansion is A329326.
Length of Lyndon factorization of reversed binary expansion is A329313.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Reversed co-necklaces are A328595.
- Length of Lyndon factorization is A329312.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Dealings are counted by A333939.
- Reversed necklaces are A333943.
- Length of co-Lyndon factorization is A334029.
- Combinatory separations are A334030.
Cf. A000740, A001037, A034691, A060223, A211100, A269134, A292884, A328596.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
colynprod[]:={};colynprod[{},b_List]:=b;colynprod[a_List,{}]:=a;colynprod[a_List]:=a;
colynprod[{x_,a___},{y_,b___}]:=Switch[Ordering[If[x=!=y,{x,y},{colynprod[{a},{x,b}],colynprod[{x,a},{b}]}]],{1,2},Prepend[colynprod[{a},{y,b}],x],{2,1},Prepend[colynprod[{x,a},{b}],y]];
colynprod[a_List,b_List,c__List]:=colynprod[a,colynprod[b,c]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
dealings[q_]:=Union[Function[ptn,Sort[q[[#]]&/@ptn]]/@sps[Range[Length[q]]]];
Table[Length[Select[dealings[stc[n]],colynprod@@#==stc[n]&]],{n,0,100}]

For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A034691(n).

A257250 Numbers n for which A256999(n) = n; numbers that cannot be made any larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 26, 28, 30, 31, 32, 48, 52, 56, 58, 60, 62, 63, 64, 96, 100, 104, 106, 112, 114, 116, 118, 120, 122, 124, 126, 127, 128, 192, 200, 208, 212, 224, 226, 228, 232, 234, 236, 240, 242, 244, 246, 248, 250, 252, 254, 255, 256, 384, 392, 400, 416, 420, 424, 426, 448, 450

Offset: 0

Antti Karttunen, May 16 2015

These correspond to the maximal (lexicographically largest) representatives selected from each equivalence class of binary necklaces. See the last example.
Indexing starts from zero, because a(0) = 0 is a special case.
If k is a member then so also is 2*k, i.e., k with 0 appended to the end of its binary representation.
If k is a member then so also is A004755(k), i.e., k with 1 prepended to the front of its binary representation.
One obtains A065609 if one erases the most significant bit of each term [as A053645(a(n))] and then discards any zero-terms produced from the terms that originally were powers of two (A000079).
First differs from A328607 in lacking 108, with binary expansion 1101100. If we define a dual-necklace to be a finite sequence that is lexicographically maximal (not minimal) among all of its cyclic rotations, these are numbers whose binary expansion, without the most significant digit, is a dual-necklace. - Gus Wiseman, Nov 04 2019

			For n = 5, with binary representation "101", if we rotate other bits than the most significant bit (that is, only the two rightmost digits "01") one step to either direction, we get "110" = 6 > 5, so 5 can be made larger by such rotations, and thus is NOT included in this sequence.
For n = 6, with binary representation "110", no such rotation will yield a larger number, and thus 6 is included in this sequence.
For n = 28, with binary representation "11100", if we rotate non-msb bits towards right, we get additional numbers 22, 19 and 25 (with binary representations "10110", "10011", "11001") before coming to 28 again, and 28 is the largest of these numbers, thus 28 is included in this sequence.
  Also, if we discard the most significant bit of each and consider them just as binary strings, then A053645(28) = 12 is the lexicographically largest representative of {"1100", "0110", "0011", "1001"}, which is the complete set of representatives for a particular equivalence class of binary necklaces, obtained by rotating all bits of binary string "1100" successively towards right or left.

Antti Karttunen, Table of n, a(n) for n = 0..16637
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Complement: A257739.
Odd terms: A000225.
Subsequence of A065609.
Cf. A004755, A053645, A080541, A080542, A256999.
Subsequence: A258003.
The non-dual version is A328668.
The version involving all digits is A065609.
The non-dual reversed version is A328607.
Numbers whose reversed binary expansion is a necklace are A328595.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000120, A001037, A003714, A014081, A069010, A275692, A328594, A328596.

reckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[0,110],#<=1||reckQ[Rest[IntegerDigits[#,2]]]&] (* Gus Wiseman, Nov 04 2019 *)

A334029 Length of the co-Lyndon factorization of the k-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 5, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 2

Offset: 0

Gus Wiseman, Apr 14 2020

nonn

We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1,0,0,1) has co-Lyndon factorization {(1),(1,0,0)}.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 441st composition in standard order is (1,2,1,1,3,1), with co-Lyndon factorization {(1),(3,1),(2,1,1)}, so a(441) = 3.

The dual version is A329312.
The version for binary expansion is (also) A329312.
The version for reversed binary expansion is A329326.
Binary Lyndon/co-Lyndon words are counted by A001037.
Necklaces covering an initial interval are A019536.
Lyndon/co-Lyndon compositions are counted by A059966
Length of Lyndon factorization of binomial expansion is A211100.
Numbers whose prime signature is a necklace are A329138.
Length of Lyndon factorization of reversed binary expansion is A329313.
A list of all binary co-Lyndon words is A329318.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Reversed co-necklaces are A328595.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
- Reversed necklaces are A333943.
- Co-necklaces are A334028.
Cf. A034691, A060223, A102659, A211097, A292884, A296372, A328596, A329358, A329359, A329362, A329400, A329401, A333939.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#1],q}]=={RotateRight[q,#1],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],colynQ[Take[q,#1]]&]]]]
Table[Length[colynfac[stc[n]]],{n,0,100}]

A334265 Numbers k such that the k-th composition in standard order is a reversed Lyndon word.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 21, 23, 32, 33, 34, 35, 37, 39, 41, 43, 47, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 135, 137, 138, 139, 141, 143, 145, 146, 147, 149, 151, 155, 159, 161, 163

Offset: 1

Gus Wiseman, Apr 22 2020

nonn

Reversed Lyndon words are different from co-Lyndon words (A326774).
A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence of all reversed Lyndon words begins:
    0: ()            37: (3,2,1)         83: (2,3,1,1)
    1: (1)           39: (3,1,1,1)       85: (2,2,2,1)
    2: (2)           41: (2,3,1)         87: (2,2,1,1,1)
    4: (3)           43: (2,2,1,1)       91: (2,1,2,1,1)
    5: (2,1)         47: (2,1,1,1,1)     95: (2,1,1,1,1,1)
    8: (4)           64: (7)            128: (8)
    9: (3,1)         65: (6,1)          129: (7,1)
   11: (2,1,1)       66: (5,2)          130: (6,2)
   16: (5)           67: (5,1,1)        131: (6,1,1)
   17: (4,1)         68: (4,3)          132: (5,3)
   18: (3,2)         69: (4,2,1)        133: (5,2,1)
   19: (3,1,1)       71: (4,1,1,1)      135: (5,1,1,1)
   21: (2,2,1)       73: (3,3,1)        137: (4,3,1)
   23: (2,1,1,1)     74: (3,2,2)        138: (4,2,2)
   32: (6)           75: (3,2,1,1)      139: (4,2,1,1)
   33: (5,1)         77: (3,1,2,1)      141: (4,1,2,1)
   34: (4,2)         79: (3,1,1,1,1)    143: (4,1,1,1,1)
   35: (4,1,1)       81: (2,4,1)        145: (3,4,1)

The non-reversed version is A275692.
The generalization to necklaces is A333943.
The dual version (reversed co-Lyndon words) is A328596.
The case that is also co-Lyndon is A334266.
Binary Lyndon words are counted by A001037.
Lyndon compositions are counted by A059966.
Normal Lyndon words are counted by A060223.
Numbers whose prime signature is a reversed Lyndon word are A334298.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Reversed Lyndon words are A334265 (this sequence).
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Distinct rotations are counted by A333632.
- Lyndon factorizations are counted by A333940.
- Length of Lyndon factorization of reverse is A334297.
Cf. A000031, A027375, A138904, A211100, A328595, A329131, A329313, A333764, A334028, A334267.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
Select[Range[0,100],lynQ[Reverse[stc[#]]]&]

A302291 a(n) is the period of the binary expansion of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 3, 1, 4, 4, 2, 4, 4, 4, 4, 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 1, 6, 6, 6, 6, 3, 6, 6, 6, 6, 6, 2, 6, 6, 3, 6, 6, 6, 6, 6, 6, 6, 6, 3, 6, 6, 6, 6, 6, 6, 6, 6, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Rémy Sigrist, Apr 04 2018

Zero is assumed to be represented as 0; otherwise, leading zeros are ignored.

See A302295 for the variant where leading zeros are allowed.

			The first terms, alongside the binary expansion of n with periodic part in parentheses, are:
  n  a(n)    bin(n)
  -- ----    ------
   0    1    (0)
   1    1    (1)
   2    2    (10)
   3    1    (1)(1)
   4    3    (100)
   5    3    (101)
   6    3    (110)
   7    1    (1)(1)(1)
   8    4    (1000)
   9    4    (1001)
  10    2    (10)(10)
  11    4    (1011)
  12    4    (1100)
  13    4    (1101)
  14    4    (1110)
  15    1    (1)(1)(1)(1)
  16    5    (10000)
  17    5    (10001)
  18    5    (10010)
  19    5    (10011)
  20    5    (10100)

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
Numbers whose prime signature is aperiodic are A329139.
Compositions by number of distinct rotations are A333941.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Reversed necklaces are A333943.
Cf. A000031, A001037, A008965, A019536, A020330, A211100, A302295, A328595, A328596, A329312, A329313, A329326.

Table[If[n==0,1,Length[Union[Array[RotateRight[IntegerDigits[n,2],#]&,IntegerLength[n,2]]]]],{n,0,50}] (* Gus Wiseman, Apr 19 2020 *)

a(n) = my (l=max(1, #binary(n))); fordiv (l, w, if (#Set(digits(n, 2^w))<=1, return (w)))

a(n) = A070939(n) / A138904(n).
a(2^n) = n + 1 for any n >= 0.
a(2^n - 1) = 1 for any n >= 0.
a(A020330(n)) = a(n) for any n > 0.

A333939 Number of multisets of compositions that can be shuffled together to obtain the k-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 2, 5, 4, 5, 1, 2, 2, 4, 2, 4, 5, 7, 2, 5, 4, 10, 4, 10, 7, 7, 1, 2, 2, 4, 2, 5, 5, 7, 2, 5, 3, 9, 5, 13, 11, 12, 2, 5, 5, 10, 5, 11, 13, 18, 4, 10, 9, 20, 7, 18, 12, 11, 1, 2, 2, 4, 2, 5, 5, 7, 2, 4, 4, 11, 5, 14, 11, 12, 2

Offset: 0

Gus Wiseman, Apr 15 2020

nonn

Number of ways to deal out the k-th composition in standard order to form a multiset of hands.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The dealings for n = 1, 3, 7, 11, 13, 23, 43:
  (1)  (11)    (111)      (211)      (121)      (2111)        (2211)
       (1)(1)  (1)(11)    (1)(21)    (1)(12)    (11)(21)      (11)(22)
               (1)(1)(1)  (2)(11)    (1)(21)    (1)(211)      (1)(221)
                          (1)(1)(2)  (2)(11)    (2)(111)      (21)(21)
                                     (1)(1)(2)  (1)(1)(21)    (2)(211)
                                                (1)(2)(11)    (1)(1)(22)
                                                (1)(1)(1)(2)  (1)(2)(21)
                                                              (2)(2)(11)
                                                              (1)(1)(2)(2)

Multisets of compositions are counted by A034691.
Combinatory separations of normal multisets are counted by A269134.
Dealings with total sum n are counted by A292884.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of Lyndon factorization of reversed binary expansion is A329313.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Aperiodic compositions are A328594.
- Length of Lyndon factorization is A329312.
- Distinct rotations are counted by A333632.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
- Length of co-Lyndon factorization is A334029.
- Combinatory separations are A334030.
Cf. A000031, A000740, A001037, A008965, A027375, A059966, A060223, A211100, A328595, A328596, A333764, A333943.

nn=100;
comps[0]:={{}};comps[n_]:=Join@@Table[Prepend[#,i]&/@comps[n-i],{i,n}];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
dealings[q_]:=Union[Function[ptn,Sort[q[[#]]&/@ptn]]/@sps[Range[Length[q]]]];
Table[Length[dealings[stc[n]]],{n,0,nn}]

For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A292884(n).

A329142 Numbers whose prime signature is not a necklace.

Original entry on oeis.org

1, 12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

After a(1) = 1, first differs from A112769 in lacking 1350.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their prime signatures begins:
   1: ()
  12: (2,1)
  20: (2,1)
  24: (3,1)
  28: (2,1)
  40: (3,1)
  44: (2,1)
  45: (2,1)
  48: (4,1)
  52: (2,1)
  56: (3,1)
  60: (2,1,1)
  63: (2,1)
  68: (2,1)
  72: (3,2)
  76: (2,1)
  80: (4,1)
  84: (2,1,1)
  88: (3,1)
  90: (1,2,1)
  92: (2,1)

Complement of A329138.
Binary necklaces are A000031.
Non-necklace compositions are A329145.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is periodic are A329140.
Cf. A001037, A008965, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329139.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[100],#==1||!neckQ[Last/@FactorInteger[#]]&]

A334266 Numbers k such that the k-th composition in standard order is both a reversed Lyndon word and a co-Lyndon word.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 21, 23, 32, 33, 34, 35, 37, 39, 43, 47, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 135, 137, 138, 139, 141, 143, 146, 147, 149, 151, 155, 159, 171, 173, 175, 183, 191

Offset: 1

Gus Wiseman, Apr 22 2020

nonn

A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations. Co-Lyndon is defined similarly, except with strictly greater instead of strictly less.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence of all reversed Lyndon co-Lyndon words begins:
    0: ()            37: (3,2,1)         91: (2,1,2,1,1)
    1: (1)           39: (3,1,1,1)       95: (2,1,1,1,1,1)
    2: (2)           43: (2,2,1,1)      128: (8)
    4: (3)           47: (2,1,1,1,1)    129: (7,1)
    5: (2,1)         64: (7)            130: (6,2)
    8: (4)           65: (6,1)          131: (6,1,1)
    9: (3,1)         66: (5,2)          132: (5,3)
   11: (2,1,1)       67: (5,1,1)        133: (5,2,1)
   16: (5)           68: (4,3)          135: (5,1,1,1)
   17: (4,1)         69: (4,2,1)        137: (4,3,1)
   18: (3,2)         71: (4,1,1,1)      138: (4,2,2)
   19: (3,1,1)       73: (3,3,1)        139: (4,2,1,1)
   21: (2,2,1)       74: (3,2,2)        141: (4,1,2,1)
   23: (2,1,1,1)     75: (3,2,1,1)      143: (4,1,1,1,1)
   32: (6)           77: (3,1,2,1)      146: (3,3,2)
   33: (5,1)         79: (3,1,1,1,1)    147: (3,3,1,1)
   34: (4,2)         85: (2,2,2,1)      149: (3,2,2,1)
   35: (4,1,1)       87: (2,2,1,1,1)    151: (3,2,1,1,1)

The version for binary expansion is A334267.
Compositions of this type are counted by A334269.
Normal sequences of this type are counted by A334270.
Necklace compositions of this type are counted by A334271.
Binary Lyndon words are counted by A001037.
Lyndon compositions are counted by A059966.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Reversed Lyndon words are A334265.
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Length of Lyndon factorization of reverse is A334297.
- Length of co-Lyndon factorization is A334029.
- Length of co-Lyndon factorization of reverse is A329313.
- Distinct rotations are counted by A333632.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
Cf. A000740, A008965, A065609, A329324, A333764, A333943, A334272, A334297.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Select[Range[0,100],lynQ[Reverse[stc[#]]]&&colynQ[stc[#]]&]

Intersection of A334265 and A326774.

cp's OEIS Frontend

A329324 Number of Lyndon compositions of n whose reverse is not a co-Lyndon composition.

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A329325 Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n with first digit removed.

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A333765 Number of co-Lyndon factorizations of the k-th composition in standard order.

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A257250 Numbers n for which A256999(n) = n; numbers that cannot be made any larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

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A334029 Length of the co-Lyndon factorization of the k-th composition in standard order.

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A334265 Numbers k such that the k-th composition in standard order is a reversed Lyndon word.

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A302291 a(n) is the period of the binary expansion of n.

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A333939 Number of multisets of compositions that can be shuffled together to obtain the k-th composition in standard order.

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